It may or may not be a big deal. The picture you're missing is the one that shows the bottom of the faucet where it comes through the sink. You definitely have or had water run down the risers and supply lines. This in turn change the patina of the copper to green. The question is, "is the water leaking from the riser to faucet connection, the faucet itself, or is it leaking from the sink surface and coming through the faucet mounting holes in the sink top?" Run the hot and cold separate, grab a flash light, stick your head under the sink and see where it's coming from.
When it's all said and done, you can cut out the old copper, water stops, escutcheons and risers, and replace them to clean things up. The bottom of the vanity can be sanded, sealed with Kilz and given a fresh coat of white paint.
You may or may not have a plumbing problem...but a chemical corrosion issue from the cleaning supplies.
I would check both the supply valves and the faucet valves for leaks. If this wer in my home I would shut off the main water and swap out the old supply vales with some new 1/4 turn ball valves. While that is being done replace the rusty escutcheon rings. Upgrading the hard supply line with some braided Stainless flex would also tidy things up.
as a follow up take a look at your cleaners...and store those properly.
Oftentimes the faucets leak at the handle when turned on. This happens also on older tub faucets as well.
Using toilet paper, wrap some around the pipe above the turn offs on the larger pipe. Turn on the faucets and let run for a short while. Then check the paper. Is it wet? If so you need to repack or change the faucets as they are leaking. If the paper gets wet before you even operate the faucet then the connection is leaking at the base of the faucet itself.
Also if that does not show the leak, put paper back on pipes, and clean the faucets as you would normally do. I expect your using a bit of water and or cleaner and spraying the valve handles etc to remove soap and water stains. If the faucet is not properly sealed to the top of the sink or counter, this happens a lot, then water is leaking around the base and that can cause this issue as well. The fix for that is simply remove the faucet and use plumbers putty or silicon caulk to seal the base.
Be sure to be liberal with the water on and around the faucet. It may not show up if your thinking your cleaning as you normally do. Also think others that use this sink may be a bit more sloppy when using it and splashing water that you would not normally do when your using it. You need to think out of the box when figuring this type of issue out. In any case if you get enough water around the sink faucets and if they are leaking at the base, you should soon see it.
it appears however that the hot water side is worse, which leads me to think it is the valve that is leaking when its on. Unless you have two separate faucets instead of a single one that has a cover over all the holes.
Going to add one more thing. It looks as though your trap is leaking also. Be sure to run hot water down the drain and using the toilet paper trick check around all the joints for leakage. Hot water will expand the pipes if leaks are going to happen, that is when it will. The reason for TP and not facial tissue, is the TP will absorb the water while the tissue will not.
When checking for sink leaks that are not easily identified, I like to use some brown paper bag, cereal box (the inside dark cardboard), or other cardboard. I will cut it to fit in the base, tightly around the pipes, and using multiple pieces if needed. Then it is easy to visibly see the water drips and where they are coming from.
I would buy TSP cleaner and any hardward store/homecenter. Use it to clean pipes which is a simple job. It appears to be a corrosive cleaning compound which caused this problem. Great you discovered before any leaks so all is good.
Thank you for all of the suggestions to fix this problem! I really appreciate it!
Thanks for the suggestion! And, you are from St. Louis. My family is from there originally with my dad's family still in the city! Thanks! laura
Thanks @Not a Trophy Wife and best of luck on the leak. Good to hear we have a local connection! St. Louis is a great town.
MIGHT OT BE A LEAK
IF THAT IS THE COLD WATER PIPE .AND I CAN SEE ITS NOT INSULATED] IT COULD JUST BE CONDENSATION